The n-Category Café

Section 4.5.3 of Marmolejo’s thesis tells us of a 2-adjunction between $PRETOP^{op}$ and $CAT$, with 2-functors $Mod(-)$ and $Set^{(-)}$. Given an algebra for the resulting 2-monad, $T$, on $CAT$, that is, a category, $A$, and functor $\Phi: Mod(Set^A) \to A$, then given an ultrafilter $(I, U)$, we can form an ultraproduct in $A$ via the ultraproduct map, $Mod(Set^A)^I \to Mod(Set^A)$.

This gives us a pre-ultracategory. Then a further 2-adjunction between $PRETOP^{op}$ and $T-ALG$ is used to generate a 2-monad whose algebras give Makkai ultracategories.

Taking our cue from Di Liberti’s Codensity: Isbell duality, pro-objects, compactness and accessibility

The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction,

I think we should look to the codensity 2-monad of the Yoneda lemma for bicategories. Kan extensions for bicategories are explained in Enriched categories as a free cocompletion, sec. 10.

Then varying the choice of 2-category in the Yoneda embedding, varying the class of limits preserved, etc. as in the ordinary case of Isbell duality should generate many 2-adjunctions along with their 2-adjoint dualities, including Gabriel-Ulmer duality and Marmolejo’s above.

James Dolan and John Baez seemed to be on this track in some notes on Doctrines:

There is a finite limits theory $Fin\Set$, which is the embodiment of propositional logic. All finite sets can be formed from $2$ by products and equalizers, so this finite limits theory is the Cauchy completion of the Lawvere theory whose objects are powers of $2$, which is the opposite of the category of finitely presented Boolean algebras. Note: $[Fin\Set, Set] = \Bool\Alg$ is the opposite of the category of profinite sets. $[Fin\Set^{op}, Set] = Set$ since $Fin\Set^{op}$ is the free finite limits theory on one object.

Categorifying this example, we should get an interesting doctrine which is the embodiment of predicate logic. Namely, let $\FP\Gpd$, the $(2,1)$-category of finitely presented groupoids. Claim: theories of this doctrine are theories of first-order predicate logic. Conjecture: $[\FP\Gpd, Gpd]$ is the opposite of the category of profinite groupoids. $[\FP\Gpd^{op}, Gpd] = Gpd$. With luck this will explain the appearance of profinite groups in number theory.

James and John work with $Gpd$ rather than $Cat$, but assuming things work out, we’d have finite limit preserving 2-functors $[\FP\Cat^{op}, Cat] = Cat$ sitting in a 2-adjunction with $[\FP\Cat, Cat]^{op}$, presumably with $Set$ acting as dualizing object, as we see with Marmolejo. Other choices would include $[\mathbf{1}, Cat] = Cat$ in duality with its opposite.

As Di Liberti shows us, pro-finite objects and compactness are often involved in one side of an Isbell duality. Then the remark about the appearance of profinite groups in number theory is intriguing. Is it that the interest in condensed/pyknotic structures arises from the nature of one side of categorified Isbell duality?

One of the instances listed in Isbell duality is that between commutative rings and affine schemes. If there is a 2-Isbell duality to be had as above, one would expect a categorification of this duality to feature. Naturally something along these lines has already appeared, see 2-algebraic geometry.

One interesting paper there is

• Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme.

There we find commutative 2-rings defined in duality with affine 2-schemes, such that any Grothendieck topos is a commutative 2-ring.

In Sec 6.2 the authors consider $C \mapsto Hom_{Com2Ring}({}^{C}Set,Set)$, for a small category $C$, where ${}^{C}Set$ is $Fun[C, Set]$. As above $Set$ is playing its role as a dualizing object. We are told this 2-functor “will be the topic of future work”. Was it, I wonder?